Fabric dependence of wave propagation in anisotropic porous media

Three-dimensional-printed replica models of bone for experimentally decoupling trabecular bone properties contribution to ultrasound propagation parameters

A detailed investigation of the relationship between ultrasonic (US) properties and trabecular bone microstructure is difficult because of the great variability in the bone loss process. The aim of this work was twofold. First, to verify by compressive tests that the three-dimensional (3D)-printer is able to produce precisely and repeatedly "bone replica models" of different size and density. Following, replicas of the original specimens with two different polymers and thinned trabeculae models were used to investigate US properties (speed of sound, SOS, and backscatter coefficient), aiming to deconvolute the influence of material properties on ultrasound characteristics. The results revealed that matrix material properties influence only the magnitude of the backscatter coefficient, whereas the characteristic undulated patterns are related to the trabecular structure. Simulation of perforation and thinning of cancellous bone, associated with bone loss, showed that SOS and mechanical properties were reduced perfectly linearly with apparent density when structure deteriorated. The 3D-printed bone replicas have the potential to enable systematic investigations of the influence of structure on both acoustical and mechanical properties and evaluate changes caused by bone loss. The development of replicas from materials with properties close to those of bone will permit quantitative conclusions for trabecular bone.

Impact behaviors of human skull sandwich cellular bones: Theoretical models and simulation

The impact behavior of human skull sandwich cellular bones with gradient geometric feature is investigated using theoretical and numerical methods. To predict the structural impact performance theoretically, the skull bone is considered as a multi-layer sandwich structure where the effect of the number of layers on its impact behavior is discussed. Three sections with different porosities and thicknesses obtained from the rebuilt 3D skull model are selected, and the numerical simulation is carried out to illustrate the reliability of the theoretical model. A close agreement between the numerical and theoretical results is observed. Moreover, the energy absorption capacity of the skull in the theoretical model is further demonstrated by experimental results of the human skull under impact loading from the literature. Numerical and experimental results show that the theoretical model can effectively predict the impact performance of the skull cellular bone. Therefore, this study can provide a reliable theoretical basis for the evaluation of the mechanical behavior of the human skull under dynamic loads.

Mechanisms of Interaction of Ultrasound With Cancellous Bone: A Review

Ultrasound is now a clinically accepted modality in the management of osteoporosis. The most common commercial clinical devices assess fracture risk from measurements of attenuation and sound speed in cancellous bone. This review discusses fundamental mechanisms underlying the interaction between ultrasound and cancellous bone. Because of its two-phase structure (mineralized trabecular network embedded in soft tissue-marrow), its anisotropy, and its inhomogeneity, cancellous bone is more difficult to characterize than most soft tissues. Experimental data for the dependencies of attenuation, sound speed, dispersion, and scattering on ultrasound frequency, bone mineral density, composition, microstructure, and mechanical properties are presented. The relative roles of absorption, scattering, and phase cancellation in determining attenuation measurements in vitro and in vivo are delineated. Common speed of sound metrics, which entail measurements of transit times of pulse leading edges (to avoid multipath interference), are greatly influenced by attenuation, dispersion, and system properties, including center frequency and bandwidth. However, a theoretical model has been shown to be effective for correction for these confounding factors in vitro and in vivo. Theoretical and phantom models are presented to elucidate why cancellous bone exhibits negative dispersion, unlike soft tissue, which exhibits positive dispersion. Signal processing methods are presented for separating "fast" and "slow" waves (predicted by poroelasticity theory and supported in cancellous bone) even when the two waves overlap in time and frequency domains. Models to explain dependencies of scattering on frequency and mean trabecular thickness are presented and compared with measurements. Anisotropy, the effect of the fluid filler medium (marrow in vivo or water in vitro), phantoms, computational modeling of ultrasound propagation, acoustic microscopy, and nonlinear properties in cancellous bone are also discussed.

ULTRASONIC WAVE IS DETERMINED BY FABRIC TENSOR: AN APPLICATION TO CALCANEUS

The fabric tensor is a good measure for determining the mechanical properties of cancellous bone. Ultrasound is one method used to measure these mechanical properties. Ultrasound-generated speed of sound (SOS) measures the mechanical properties of cancellous bone. Thus, in this paper, we started with the fact that the fast wave in poroelastic theory is identical to the bulk wave velocity. We then formulate the equation for the fast wave in terms of fabric tensor for the calcaneus. The formulation in this paper is simpler than previously published results and will be easy to use in future experiments.

Enhanced correlation between quantitative ultrasound and structural and mechanical properties of bone using combined transmission-reflection measurement

Quantitative ultrasound (QUS) is capable of predicting the principal structural orientation of trabecular bone; this orientation is highly correlated with the mechanical strength of trabecular bone. Irregular shape of bone, however, would increase variation in such a prediction, especially under human in vivo measurement. This study was designed to combine transmission and reflection modes of QUS measurement to improve the prediction for the structural and mechanical properties of trabecular bone. QUS, mechanical testing, and micro computed tomography (μCT) scanning were performed on 24 trabecular bone cubes harvested from a bovine distal femur to obtain the mechanical and structural parameters. Transmission and reflection modes of QUS measurement in the transverse and frontal planes were performed in a confined 60° angle range with 5° increment. The QUS parameters, attenuation (ATT) and velocity (UV), obtained from transmission mode, were normalized to the specimen thickness acquired from reflection mode. Analysis of covariance showed that the combined transmission-reflection modes improved prediction for the structural and Young's modulus of bone in comparison to the traditional QUS measurement performed only in the medial-lateral orientation. In the transverse plane, significant improvement between QUS and μCT was found in ATT vs bone surface density (BS/BV) (p < 0.05), ATT vs trabecular thickness (Tb.Th) (p < 0.01), ATT vs degree of anisotropy (DA) (p < 0.05), UV vs trabecular bone number (Tb.N) (p < 0.05), and UV vs Tb.Th (p < 0.001). In the frontal plane, significant improvement was found in ATT vs structural model index (SMI) (p < 0.01), ATT vs bone volume fraction (BV/TV) (p < 0.01), ATT vs BS/BV (p < 0.001), ATT vs Tb.Th (p < 0.001), ATT vs DA (p < 0.001), and ATT vs modulus (p < 0.001), UV vs SMI (p < 0.01), UV vs BV/TV (p < 0.05), UV vs BS/BV (p < 0.05), UV vs Tb.Th (p < 0.01), UV vs trabecular spacing (p < 0.05), and UV vs modulus (p < 0.01). These data suggested that the combined transmission-reflection QUS method is capable of providing information more relevant to the structural and mechanical properties of trabecular bone.

Changes of elastic constants and anisotropy patterns in trabecular bone during disuse-induced bone loss assessed by poroelastic ultrasound

Currently, the approach most widely used to examine bone loss is the measurement of bone mineral density (BMD) using dual X-ray absorptiometry (DXA). However, bone loss due to immobilization creates changes in bone microarchitecture, which in turn are related to changes in bone mechanical function and competence to resist fracture.Unfortunately, the relationship between microarchitecture and mechanical function within the framework of immobilization and antiresorptive therapy has not being fully investigated. The goal of the present study was to investigate the structure–function relationship in trabecular bone in the real-world situations of a rapidly evolving osteoporosis(disuse), both with and without antiresorptive treatment. We evaluated the structure–function relationship in trabecular bone after bone loss (disuse-induced osteoporosis)and bisphosphonate treatment (antiresorptive therapy using risedronate) in canine trabecular bone using lCT and ultrasound wave propagation. Microstructure values determined from lCT images were used into the anisotropic poroelastic model of wave propagation in order to compute the apparent elastic constants (EC) and elastic anisotropy pattern of bone. Immobilization resulted in a significant reduction in trabecular thickness (Tb.Th) and bone volume fraction (BV/TV), while risedronate treatment combined with immobilization exhibited a lesser reduction in Tb.Th and BV/TV, suggesting that risedronate treatment decelerates bone loss, but it was unable to fully stop it. Risedronate treatment also increased the tissue mineral density (TMD), which when combined with the decrease in Tb.Th and BV/TV may explain the lack of significant differences invBMD in both immobilization and risedronate treated groups. Interestingly, changes inapparent EC were much stronger in the superior–inferior (SI) direction than in the medial–lateral (ML) and anterior–posterior (AP) anatomical directions, producing changes in elastic anisotropy patterns. When data were pooled together, vBMD was able to explain 58% of ultrasound measurements variability, a poroelastic wave propagation analytical model (i.e., BMD modulated by fabric directionality) was able to predict 81%of experimental wave velocity variability, and also explained 91% of apparent EC and changes in elastic anisotropy patterns. Overall, measurements of vBMD were unable to distinguish changes in apparent EC due to immobilization or risedronate treatment.However, anisotropic poroelastic ultrasound (PEUS) wave propagation was able to distinguish functional changes in apparent EC and elastic anisotropy patterns due to immobilization and antiresorptive therapy, providing an enhanced discrimination of anisotropic bone loss and the structure–function relationship in immobilized and risedronate-treated bone, beyond vBMD.

Material Properties of the Mandibular Trabecular Bone

The present paper introduces a numerical simulation aided, experimental method for the measurement of Young's modulus of the trabecular substance in the human mandible. Compression tests were performed on fresh cadaveric samples containing trabecular bone covered with cortical layer, thus avoiding the destruction caused by the sterilization, preservation, and storage and the underestimation of the stiffness resulting from the individual failure of the trabeculae cut on the surfaces. The elastic modulus of the spongiosa was determined by the numerical simulation of each compression test using a specimen specific finite element model of each sample. The received mandibular trabecular bone Young's modulus values ranged from 6.9 to 199.5 MPa.

Multichannel instantaneous frequency analysis of ultrasound propagating in cancellous bone

An ultrasonic pulse propagating in cancellous bone can be separated into two waves depending on the condition of the specimen. These two waves, which are called the fast wave and the slow wave, provide important information for the diagnosis of osteoporosis. The present study proposes to utilize a signal processing method that extracts the instantaneous frequency (IF) of waveforms from multiple spectral channels. The instantaneous frequency was expected to be able to show detailed time-frequency properties of ultrasonic waves being transmitted through cancellous bone. The employed method, termed the multichannel instantaneous frequency (MCIF) method, showed robustness against background noise as compared to the IF that was directly derived from the original waveform. The extracted IF revealed that the frequency of the fast wave was affected by both the propagation distance within the specimen and the bone density, independently. On the other hand, the alternation of the center frequency of the originally transmitted wave did not produce proportional changes in the extracted IF values of the fast waves, suggesting that the fast wave IF mainly reflected the thickness of the specimens. These findings may provide the possibility of obtaining a more precise diagnosis of osteoporosis.

Application of acoustic bessel beams for handling of hollow porous spheres

Acoustic manipulation of porous spherical shells, widely used as drug delivery carriers and magnetic resonance imaging contrast agents, is investigated analytically. The technique used for this purpose is based on the application of high-order Bessel beams as a single-beam acoustic manipulation device, by which particles lying on the axis of the beam can be pulled toward the beam source. The exerted acoustic radiation force is calculated using the standard partial-wave series method, and the wave propagation within the porous media is modeled using Biot's theory of poro-elasticity. Numerical simulations are performed for porous aluminum and silica shells of different thickness and porosity. Results indicate that manipulation of low-porosity shells is possible using Bessel beams with large conical angles, over a number of broadband frequency ranges, whereas manipulation of highly porous shells can occur over both narrowband and broadband frequency domains.

Principal trabecular structural orientation predicted by quantitative ultrasound is strongly correlated with μFEA determined anisotropic apparent stiffness

The microarchitecture and alignment of trabecular bone adapts to the particular mechanical milieu applied to it. Due to this anisotropic mechanical property, measurement orientation has to be taken into consideration when assessing trabecular bone quality and fracture risk prediction. Quantitative ultrasound (QUS) has demonstrated the ability in predicting the principal structural orientation (PSO) of trabecular bone. Although the QUS prediction for PSO is very close to that of μCT, certain angle differences still exist. It remains unknown whether this angle difference can induce significant differences in mechanical properties or not. The objective of this study was to evaluate the mechanical properties in different PSOs predicted using different methods, QUS and μCT, thus to investigate the ability of QUS as a means to predict the PSO of trabecular bone noninvasively. By validating the ability of QUS to predict the PSO of trabecular bone, it is beneficial for future QUS applications because QUS measurements in the PSO can provide information more correlated with the mechanical properties than with other orientations. In this study, seven trabecular bone balls from distal bovine femurs were used to generate finite element models based on the 3-dimensional μCT images. Uniaxial compressive loading was performed on the bone ball models in the finite element analysis (FEA) in six different orientations (three anatomical orientations, two PSOs predicted by QUS and the longest vector of mean intercept length (MIL) tensor calculated by μCT). The stiffness was calculated based on the reaction force of the bone balls under loading, and the von Mises stress results showed that both the mechanical properties in the PSOs predicted by QUS are significantly higher than the anatomical orientations and comparatively close to the longest vector of MIL tensor. The stiffness in the PSOs predicted by QUS is also highly correlated with the stiffness in the MIL tensor orientation (ATTmax vs. MIL, R(2) = 0.98, p < 001; UVmax vs. MIL, R(2) = 0.92, p < 001). These results were validated by in vitro mechanical testing on the bone ball samples. This study demonstrates that the PSO of trabecular bone predicted by QUS has an equally strong apparent stiffness with the orientation predicted by μCT.

Estimation of fast and slow wave properties in cancellous bone using Prony's method and curve fitting

The presence of two longitudinal waves in poroelastic media is predicted by Biot's theory and has been confirmed experimentally in through-transmission measurements in cancellous bone. Estimation of attenuation coefficients and velocities of the two waves is challenging when the two waves overlap in time. The modified least squares Prony's (MLSP) method in conjuction with curve-fitting (MLSP + CF) is tested using simulations based on published values for fast and slow wave attenuation coefficients and velocities in cancellous bone from several studies in bovine femur, human femur, and human calcaneus. The search algorithm is accelerated by exploiting correlations among search parameters. The performance of the algorithm is evaluated as a function of signal-to-noise ratio (SNR). For a typical experimental SNR (40 dB), the root-mean-square errors (RMSEs) for one example (human femur) with fast and slow waves separated by approximately half of a pulse duration were 1 m/s (slow wave velocity), 4 m/s (fast wave velocity), 0.4 dB/cm MHz (slow wave attenuation slope), and 1.7 dB/cm MHz (fast wave attenuation slope). The MLSP + CF method is fast (requiring less than 2 s at SNR = 40 dB on a consumer-grade notebook computer) and is flexible with respect to the functional form of the parametric model for the transmission coefficient. The MLSP + CF method provides sufficient accuracy and precision for many applications such that experimental error is a greater limiting factor than estimation error.

Longitudinal elastic properties and porosity of cortical bone tissue vary with age in human proximal femur

Tissue level structural and mechanical properties are important determinants of bone strength. As an individual ages, microstructural changes occur in bone, e.g., trabeculae and cortex become thinner and porosity increases. However, it is not known how the elastic properties of bone change during aging. Bone tissue may lose its elasticity and become more brittle and prone to fractures as it ages. In the present study the age-dependent variation in the spatial distributions of microstructural and microelastic properties of the human femoral neck and shaft were evaluated by using acoustic microscopy. Although these properties may not be directly measured in vivo, there is a major interest to investigate their relationships with the linear elastic measurements obtained by diagnostic ultrasound at the most severe fracture sites, e.g., the femoral neck. However, before the validity of novel in vivo techniques can be established, it is essential to understand the age-dependent variation in tissue elastic properties and porosity at different skeletal sites. A total of 42 transverse cross-sectional bone samples were obtained from the femoral neck (Fn) and proximal femoral shaft (Ps) of 21 men (mean±SD age 47.1±17.8, range 17-82years). Samples were quantitatively imaged using a scanning acoustic microscope (SAM) equipped with a 50MHz ultrasound transducer. Distributions of the elastic coefficient (c33) of cortical (Ct) and trabecular (Tr) tissues and microstructure of cortex (cortical thickness Ct.Th and porosity Ct.Po) were determined. Variations in c33 were observed with respect to tissue type (c33Tr<c33Ct), location (c33(Ct.Ps)=37.7GPa>c33(Ct.Fn)=35.3GPa>c33(Tr.Ps)=33.8GPa>c33(Tr.Fn)=31.9GPa), and cadaver age (R(2)=0.28-0.46, p<0.05). Regional variations in porosity were found in the neck (superior 13.1%; inferior 6.1%; anterior 10.1%; posterior 8.6%) and in the shaft (medial 9.5%; lateral 7.7%; anterior 8.6%; posterior 12.0%). In conclusion, significant variations in elastic coefficients were detected between femoral neck and shaft as well as between the quadrants of the cross-sections of neck and shaft. Moreover, an age-related increase in cortical porosity and a stiffening of the bone tissue were observed. These findings may explain in part the increase in susceptibility to suffer low energy fractures during aging and highlight the potential of ultrasound in clinical osteoporosis diagnostics.

Tortuosity and the Averaging of Microvelocity Fields in Poroelasticity

The relationship between the macro- and microvelocity fields in a poroelastic representative volume element (RVE) has not being fully investigated. This relationship is considered to be a function of the tortuosity: a quantitative measure of the effect of the deviation of the pore fluid streamlines from straight (not tortuous) paths in fluid-saturated porous media. There are different expressions for tortuosity based on the deviation from straight pores, harmonic wave excitation, or from a kinetic energy loss analysis. The objective of the work presented is to determine the best expression for tortuosity of a multiply interconnected open pore architecture in an anisotropic porous media. The procedures for averaging the pore microvelocity over the RVE of poroelastic media by Coussy and by Biot were reviewed as part of this study, and the significant connection between these two procedures was established. Success was achieved in identifying the Coussy kinetic energy loss in the pore fluid approach as the most attractive expression for the tortuosity of porous media based on pore fluid viscosity, porosity, and the pore architecture. The fabric tensor, a 3D measure of the architecture of pore structure, was introduced in the expression of the tortuosity tensor for anisotropic porous media. Practical considerations for the measurement of the key parameters in the models of Coussy and Biot are discussed. In this study, we used cancellous bone as an example of interconnected pores and as a motivator for this study, but the results achieved are much more general and have a far broader application than just to cancellous bone.

Microarchitecture and bone quality in the human calcaneus: local variations of fabric anisotropy

The local variability of microarchitecture of human trabecular calcaneus bone is investigated using high-resolution micro-computed tomography (µCT) scanning. The fabric tensor is employed as the measure of the microarchitecture of the pore structure of a porous medium. It is hypothesized that a fabric tensor-dependent poroelastic ultrasound approach will more effectively predict the data variance than will porosity alone. The specific aims of the present study are as follows: (1) to quantify the morphology and local anisotropy of the calcaneus microarchitecture with respect to anatomical directions; (2) to determine the interdependence, or lack thereof, of microarchitecture parameters, fabric, and volumetric bone mineral density (vBMD); and (3) to determine the relative ability of vBMD and fabric measurements in evaluating the variance in ultrasound wave velocity measurements along orthogonal directions in the human calcaneus. Our results show that the microarchitecture in the analyzed regions of human calcanei is anisotropic, with a preferred alignment along the posterior-anterior direction. Strong correlation was found between most scalar architectural parameters and vBMD. However, no statistical correlation was found between vBMD and the fabric components, the measures of the pore microstructure orientation. Therefore, among the parameters usually considered for cancellous bone (ie, classic histomorphometric parameters such as porosity, trabecular thickness, number and separation), only fabric components explain the data variance that cannot be explained by vBMD, a global mass measurement, which lacks the sensitivity and selectivity to distinguish osteoporotic from healthy subjects because it is insensitive to directional changes in bone architecture. This study demonstrates that a multidirectional, fabric-dependent poroelastic ultrasound approach has the capability of characterizing anisotropic bone properties (bone quality) beyond bone mass, and could help to better understand anisotropic changes in bone architecture using ultrasound.

In vitro estimation of fast and slow wave parameters of thin trabecular bone using space-alternating generalized expectation-maximization algorithm

In testing cancellous bone using ultrasound, two types of longitudinal Biot's waves are observed in the received signal. These are known as fast and slow waves and their appearance depend on the alignment of bone trabeculae in the propagation path and the thickness of the specimen under test (SUT). They can be used as an effective tool for the diagnosis of osteoporosis because wave propagation behavior depends on the bone structure. However, the identification of these waves in the received signal can be difficult to achieve. In this study, ultrasonic wave propagation in a 4mm thick bovine cancellous bone in the direction parallel to the trabecular alignment is considered. The observed Biot's fast and slow longitudinal waves are superimposed; which makes it difficult to extract any information from the received signal. These two waves can be separated using the space alternating generalized expectation maximization (SAGE) algorithm. The latter has been used mainly in speech processing. In this new approach, parameters such as, arrival time, center frequency, bandwidth, amplitude, phase and velocity of each wave are estimated. The B-Scan images and its associated A-scans obtained through simulations using Biot's finite-difference time-domain (FDTD) method are validated experimentally using a thin bone sample obtained from the femoral-head of a 30 months old bovine.

Prediction of trabecular bone principal structural orientation using quantitative ultrasound scanning

Bone has the ability to adapt its structure in response to the mechanical environment as defined as Wolff's Law. The alignment of trabecular structure is intended to adapt to the particular mechanical milieu applied to it. Due to the absence of normal mechanical loading, it will be extremely important to assess the anisotropic deterioration of bone during the extreme conditions, i.e., long term space mission and disease orientated disuse, to predict risk of fractures. The propagation of ultrasound wave in trabecular bone is substantially influenced by the anisotropy of the trabecular structure. Previous studies have shown that both ultrasound velocity and amplitude is dependent on the incident angle of the ultrasound signal into the bone sample. In this work, seven bovine trabecular bone balls were used for rotational ultrasound measurement around three anatomical axes to elucidate the ability of ultrasound to identify trabecular orientation. Both ultrasound attenuation (ATT) and fast wave velocity (UV) were used to calculate the principal orientation of the trabecular bone. By comparing to the mean intercept length (MIL) tensor obtained from μCT, the angle difference of the prediction by UV was 4.45°, while it resulted in 11.67° angle difference between direction predicted by μCT and the prediction by ATT. This result demonstrates the ability of ultrasound as a non-invasive measurement tool for the principal structural orientation of the trabecular bone.

Mixture theory-based poroelasticity as a model of interstitial tissue growth

This contribution presents an alternative approach to mixture theory-based poroelasticity by transferring some poroelastic concepts developed by Maurice Biot to mixture theory. These concepts are a larger RVE and the subRVE-RVE velocity average tensor, which Biot called the micro-macro velocity average tensor. This velocity average tensor is assumed here to depend upon the pore structure fabric. The formulation of mixture theory presented is directed toward the modeling of interstitial growth, that is to say changing mass and changing density of an organism. Traditional mixture theory considers constituents to be open systems, but the entire mixture is a closed system. In this development the mixture is also considered to be an open system as an alternative method of modeling growth. Growth is slow and accelerations are neglected in the applications. The velocity of a solid constituent is employed as the main reference velocity in preference to the mean velocity concept from the original formulation of mixture theory. The standard development of statements of the conservation principles and entropy inequality employed in mixture theory are modified to account for these kinematic changes and to allow for supplies of mass, momentum and energy to each constituent and to the mixture as a whole. The objective is to establish a basis for the development of constitutive equations for growth of tissues.

Role of structural anisotropy of biological tissues in poroelastic wave propagation

Ultrasound waves have a broad range of clinical applications as a non-destructive testing approach in imaging and in the diagnoses of medical conditions. Generally, biological tissues are modeled as an homogenized equivalent medium with an apparent density through which a single wave propagates. Only the first wave arriving at the ultrasound probe is used for the measurement of the speed of sound. However, the existence of a second wave in tissues such as cancellous bone has been reported and its existence is an unequivocal signature of Biot type poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as density, a fabric-dependent anisotropic poroelastic ultrasound (PEU) propagation theory was recently developed. Key to this development was the inclusion of the fabric tensor - a quantitative stereological measure of the degree of structural anisotropy of bone - into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of waves in several soft and hard tissues. It was found that collagen fibers in soft tissues and the mineralized matrix in hard tissues are responsible for the anisotropy of the solid tissue constituent through the fabric tensor in the model.

Stochastic theory of dynamic permeability in poroelastic media

A theory for the dynamic permeability in deformable porous media is developed. The analysis is based on the momentum flux transfer from the slow compressional into the slow shear wave (a proxy for the viscous wave in a Newtonian fluid) in the presence of random pore-scale heterogeneities. A first-order statistical smoothing approximation is used to infer a dynamic permeability in the form of an integral over the covariance function modulated by the slow shear wave. In a smooth pore-throat limit the results reproduce the model proposed by Johnson et al. [J. Fluid Mech. 176, 379 (1987)].

The specific growth rates of tissues: a review and a re-evaluation

The first objective of this review and re-evaluation is to present a brief history of efforts to mathematically model the growth of tissues. The second objective is to place this historical material in a current perspective where it may be of help in future research. The overall objective is to look backward in order to see ways forward. It is noted that two distinct methods of imaging or modeling the growth of an organism were inspired over 70 years ago by Thompson's (1915, "XXVII Morphology and Mathematics," Trans. - R. Soc. Edinbrgh, 50, pp. 857-895; 1942, On Growth and Form, Cambridge University Press, Cambridge, UK) method of coordinate transformations to study the growth and form of organisms. One is based on the solid mechanics concept of the deformation of an object, and the other is based on the fluid mechanics concept of the velocity field of a fluid. The solid mechanics model is called the distributed continuous growth (DCG) model by Skalak (1981, "Growth as a Finite Displacement Field," Proceedings of the IUTAM Symposium on Finite Elasticity, D. E. Carlson and R. T. Shield, eds., Nijhoff, The Hague, pp. 348-355) and Skalak et al. (1982, "Analytical Description of Growth," J. Theor. Biol., 94, pp. 555-577), and the fluid mechanics model is called the graphical growth velocity field representation (GVFR) by Cowin (2010, "Continuum Kinematical Modeling of Mass Increasing Biological Growth," Int. J. Eng. Sci., 48, pp. 1137-1145). The GVFR is a minimum or simple model based only on the assumption that a velocity field may be used effectively to illustrate experimental results concerning the temporal evolution of the size and shape of the organism that reveals the centers of growth and growth gradients first described by Huxley (1924, "Constant Differential Growth-Ratios and Their Significance," Nature (London), 114, pp. 895-896; 1972, Problems of Relative Growth, 2nd ed., L. MacVeagh, ed., Dover, New York). It is the method with an independent future that some earlier writers considered as an aspect of the DCG model. The development of the DCG hypothesis and the mixture theory models into models for the predicted growth of an organism is taking longer because these models are complicated and the development and refinement of the basic concepts are slower.

Fabric dependence of quasi-waves in anisotropic porous media

Assessment of bone loss and osteoporosis by ultrasound systems is based on the speed of sound and broadband ultrasound attenuation of a single wave. However, the existence of a second wave in cancellous bone has been reported and its existence is an unequivocal signature of poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as bone mineral density (BMD), a fabric-dependent anisotropic poroelastic wave propagation theory was recently developed for pure wave modes propagating along a plane of symmetry in an anisotropic medium. Key to this development was the inclusion of the fabric tensor--a quantitative stereological measure of the degree of structural anisotropy of bone--into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of mixed wave modes along an arbitrary direction in anisotropic porous media called quasi-waves. It was found that differences between phase and group velocities are due to the anisotropy of the bone microarchitecture, and that the experimental wave velocities are more accurately predicted by the poroelastic model when the fabric tensor variable is taken into account. This poroelastic wave propagation theory represents an alternative for bone quality assessment beyond BMD.

Cancellous bone analysis with modified least squares Prony's method and chirp filter: phantom experiments and simulation

The presence of two longitudinal waves in porous media is predicted by Biot's theory and has been confirmed experimentally in cancellous bone. When cancellous bone samples are interrogated in through-transmission, these two waves can overlap in time. Previously, the Modified Least-Squares Prony's (MLSP) method was validated for estimation of amplitudes, attenuation coefficients, and phase velocities of fast and slow waves, but tended to overestimate phase velocities by up to about 5%. In the present paper, a pre-processing chirp filter to mitigate the phase velocity bias is derived. The MLSP/chirp filter (MLSPCF) method was tested for decomposition of a 500 kHz-center-frequency signal containing two overlapping components: one passing through a low-density-polyethylene plate (fast wave) and another passing through a cancellous-bone-mimicking phantom material (slow wave). The chirp filter reduced phase velocity bias from 100 m/s (5.1%) to 69 m/s (3.5%) (fast wave) and from 29 m/s (1.9%) to 10 m/s (0.7%) (slow wave). Similar improvements were found for 1) measurements in polycarbonate (fast wave) and a cancellous-bone-mimicking phantom (slow wave), and 2) a simulation based on parameters mimicking bovine cancellous bone. The MLSPCF method did not offer consistent improvement in estimates of attenuation coefficient or amplitude.